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Consider the sum, Z, of two independent binomial random variables, X ~ B(m 0, p 0) and Y ~ B(m 1, p 1), where Z = X + Y.Then, the variance of Z is less than or equal to its variance under the assumption that p 0 = p 1 = ¯, that is, if Z had a binomial distribution with the success probability equal to the average of X and Y 's probabilities. [8]
Median: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half. Mode: for a discrete random variable, the value with highest probability; for an absolutely continuous random variable, a location at which the probability density function has a local peak.
The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables. [citation needed] One author uses the terminology of the "Rule of Average Conditional Probabilities", [4] while another refers to it as the "continuous law of ...
The probabilities of rolling several numbers using two dice. Probability is the branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur.
This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). [1]
Since a Poisson binomial distributed variable is a sum of n independent Bernoulli distributed variables, its mean and variance will simply be sums of the mean and variance of the n Bernoulli distributions: = =
The product is one type of algebra for random variables: Related to the product distribution are the ratio distribution, sum distribution (see List of convolutions of probability distributions) and difference distribution. More generally, one may talk of combinations of sums, differences, products and ratios.
In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. This inequality provides an upper bound on the probability of occurrence of at least one ...